Chapter 454

Kaehler's Theory of Hyperpurity on Manifolds and Clifford's Analysis

This is the name of Cheng Nuo's graduation thesis.

In the past six months, the geometry of complex manifolds has once again become one of the popular research directions.

The Kaehler manifold is a complex manifold with a Riemannian metric that does not change under the action of a typical complex structure.

The typical complex structure of the Kaehler manifold is parallel under the corresponding Riemann contact. Therefore, Kaehler manifolds are a special class of Riemannian manifolds with richer geometric structures and thus more colorful geometric properties.

In addition, Kaehler manifolds can also be studied from the perspective of algebraic geometry, and the geometry of Kaehler manifolds can be explained by differential geometry.

Overall, the Kaehler manifold is an object of study that encompasses almost all branches of geometry.

This is also an important reason why Cheng Nuo decided that the Kaehler manifold was the topic of his graduation thesis.

............

Outside the world, the news about Cheng Nuo's proof of Jacobi's conjecture is still fermenting.

At the age of 21, he is a prover of conjecture, which has attracted the attention of almost all major mathematics universities around the world.

The 21-year-old guys at their school had just graduated from undergrad.

And Cheng Nuo, at the same age, has already proved the Jacobian conjecture, one of the major mathematical conjectures in the geometric world.

This makes the universities feel embarrassed.

They thought that Cheng Nuo, a young man, had been silent for half a year, but now he finally made a big move, and he would make a wave of high-profile publicity.

Outcome...... Not really.

Except for the news of Cheng Nuo's return to MIT, nearly a month has passed, and Cheng Nuo seems to have evaporated from the world, and it is difficult to find his traces.

"Hoo ——!"

MIT Library, a location near the window.

The warm sun shone on Cheng Nuo, and he stretched his waist and looked at the results of his struggle for a month with satisfaction.

[Massachusetts Institute of Technology

Master's thesis

Ultra-pure theory and Clifford analysis on Kaehler manifolds

Name: Cheng Nuo

Major: Basic Mathematics

Instructor: Fresnel Doyle

Abstract:There are many differences between the theory of multiple complex variable functions and the theory of single complex variable functions, for example, there is the famous Hartogs phenomenon in multiple complex variables, but not in single complex variables, and the famous Riemarm mapping fundamental theorem is no longer true in the multi-complex variable space......

Cheng Nuo checked his dissertation again from beginning to end, simplified the derivation process in a few places, and then sent the 32-page dissertation to Professor Fresnel.

Cheng Nuo: "Professor, I've finished writing my graduation thesis. ”

Professor Fresnel: "Well, I'll let you know when I'll defend my graduation later." ”

More than half an hour later, Professor Fresnel sent a message to Cheng Nuo.

Professor Fresnel, "Next Wednesday at 10 a.m., the lecture hall on the first floor of the School of Mathematics will be used for the graduation defense, so don't be late. ”

Cheng Nuo: "I know, I will be there on time." ”

............

Regarding the graduation defense, Cheng Nuo does not need to do much preparation.

In fact, for MIT, Cheng Nuo's graduation defense is just a process.

If a ruthless person who even proves Jacobi's conjecture can't even pass the graduation defense, people from the outside world are afraid that there will be some conspiracy theory about MIT.

Wednesday, a very ordinary day.

Cheng Nuo changed into a suit and arrived at the lecture hall on time before the defense began.

This graduation defense was specially prepared for Cheng Nuo.

It's October, not June's graduation season.

Therefore, the four teachers of the defense team all serve Cheng Nuo.

This is not a precedent.

But this kind of guy who forced the school to graduate can't produce two or three in ten years on average.

There are four members of the defense team, and the leader is naturally Professor Fresnel. The other three are all professors of geometry at the school.

Although they just walked through the process, the four of them did not intend to let go of the process easily.

In particular, the three professors of the geometry world are constantly frantically testing the strength of this newcomer who is known as the strongest newcomer in the geometry world.

In the questioning session, Cheng Nuo didn't have time to rest at all, and kept accepting questions from the three professors one after another.

"Cheng Nuo, on page 11 of your paper, in the Kaehler manifold superpure D problem, you use matrix differentiation to define the hyperpure Cauchy-Riemann operator, what is the purpose?"

"First of all, the elements of a matrix include not only the differential form, but also the so-called shrinkage operator. Let's say that in {1... ,n) and they are r-multivariate group j=(j1,...... jr) and ......"

"In complex Clifford algebras, is there any other way to classify complex algebras produced in complex differential form and complex algebra produced by operator dz into the same algebra than the complex algebra that uses the multiplication rule mentioned in your paper?"

"This ......" Cheng Nuo touched his chin and thought for a few seconds, "This one is there." ”

"You can use the formula, e^2=-1=ae0, k∈{1,... ，2n}，ekeq+eqek=0，k≠q，......”

............

Originally planned for a half-hour question session, Cheng Nuo and the three professors directly exchanged questions and answers for nearly an hour.

Cheng Nuo's mouth was dry, and his mouth almost never stopped during the period.

Fortunately, the three professors who found that they couldn't ask Cheng Nuo no longer continued to make things difficult for Cheng Nuo and ended this link.

Ten minutes later......

Professor Fresnel sent the diploma and degree certificate that had been stamped in advance to Cheng Nuo. "Cheng Nuo, on behalf of the Massachusetts Institute of Technology, I officially confer on you a Doctor of Science degree!"

Doctor of Science!

This is the highest degree in MIT's science major, and it is an unattainable existence that thousands of students have struggled for years.

Cheng Nuo, on the other hand, easily got his Ph.D. after one year of study at MIT.

It's a Doctor of Science, not a Master of Science.

Cheng Nuo was a little surprised.

He directly skipped the doctoral stage and got a doctorate.

Professor Fresnel took the medal of Doctor of Science, put it on Cheng Nuo's neck, and patted the shoulder that was still stunned, "This is what you deserve." ”

"Not to mention your previous achievements, just your graduation thesis, in terms of academic level, has long exceeded the requirements for doctoral graduation!"

"I have contacted one of the editors-in-chief of Mathematical Symbols of the United States, and your dissertation will be published in the next issue of the journal. ”

Although the status of "Mathematical Symbols" in the mathematics community of the United States is not as good as that of "Annals of Mathematics", it is no problem to rank second, and it is also one of the top journals.

The next day, Cheng Nuo wore a degree gown and took a graduation photo in front of the library.

Toss the Doctor's hat high into the sky and watch as it rapidly falls from its freefall.

Cheng Nuo suddenly realized that his student days were over!

..................